<!DOCTYPE html>
<html lang="en">

<head>
  <meta charset="utf-8" />
   
  <meta name="keywords" content="Java Python" />
   
  <meta name="description" content="From Zero to Hero" />
  
  <meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1" />
  <title>
    Leetcode-202-快乐数 |  朱酱酱的学习博客
  </title>
  <meta name="generator" content="hexo-theme-yilia-plus">
  
  <link rel="shortcut icon" href="/favicon.ico" />
  
  
<link rel="stylesheet" href="/css/style.css">

  
<script src="/js/pace.min.js"></script>


  

  

<link rel="alternate" href="/atom.xml" title="朱酱酱的学习博客" type="application/atom+xml">
</head>

</html>

<body>
  <div id="app">
    <main class="content">
      <section class="outer">
  <article id="post-Leetcode/Leetcode-202-快乐数" class="article article-type-post" itemscope
  itemprop="blogPost" data-scroll-reveal>

  <div class="article-inner">
    
    <header class="article-header">
       
<h1 class="article-title sea-center" style="border-left:0" itemprop="name">
  Leetcode-202-快乐数
</h1>
  

    </header>
    

    
    <div class="article-meta">
      <a href="/2020/06/08/Leetcode/Leetcode-202-%E5%BF%AB%E4%B9%90%E6%95%B0/" class="article-date">
  <time datetime="2020-06-08T01:12:53.000Z" itemprop="datePublished">2020-06-08</time>
</a>
      
      
      
<div class="word_count">
    <span class="post-time">
        <span class="post-meta-item-icon">
            <i class="ri-quill-pen-line"></i>
            <span class="post-meta-item-text"> 字数统计:</span>
            <span class="post-count">1.6k字</span>
        </span>
    </span>

    <span class="post-time">
        &nbsp; | &nbsp;
        <span class="post-meta-item-icon">
            <i class="ri-book-open-line"></i>
            <span class="post-meta-item-text"> 阅读时长≈</span>
            <span class="post-count">6分钟</span>
        </span>
    </span>
</div>

      
    </div>
    

    
    
    <div class="tocbot"></div>





    

    <div class="article-entry" itemprop="articleBody">
      


      

      
      <h1 id="Leetcode-202-Happy-Number"><a href="#Leetcode-202-Happy-Number" class="headerlink" title="Leetcode-202-Happy Number"></a>Leetcode-202-<a href="https://leetcode-cn.com/problems/happy-number/" target="_blank" rel="noopener">Happy Number</a></h1><h2 id="思路："><a href="#思路：" class="headerlink" title="思路："></a>思路：</h2><h2 id="题目描述"><a href="#题目描述" class="headerlink" title="题目描述"></a>题目描述</h2><p>编写一个算法来判断一个数 n 是不是快乐数。</p>
<p>「快乐数」定义为：对于一个正整数，每一次将该数替换为它每个位置上的数字的平方和，然后重复这个过程直到这个数变为 1，也可能是 无限循环 但始终变不到 1。如果 可以变为  1，那么这个数就是快乐数。</p>
<p>如果 n 是快乐数就返回 True ；不是，则返回 False 。</p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line">输入：19</span><br><span class="line">输出：true</span><br><span class="line">解释：</span><br><span class="line">12 + 92 &#x3D; 82</span><br><span class="line">82 + 22 &#x3D; 68</span><br><span class="line">62 + 82 &#x3D; 100</span><br><span class="line">12 + 02 + 02 &#x3D; 1</span><br></pre></td></tr></table></figure>



<a id="more"></a>



<p><strong>参考题解</strong>：<a href="https://leetcode-cn.com/problems/happy-number/solution/kuai-le-shu-by-leetcode-solution/" target="_blank" rel="noopener">https://leetcode-cn.com/problems/happy-number/solution/kuai-le-shu-by-leetcode-solution/</a></p>
<h2 id="方法一：哈希表"><a href="#方法一：哈希表" class="headerlink" title="方法一：哈希表"></a>方法一：哈希表</h2><figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">我们可以先举几个例子。我们从 7 开始。则下一个数字是 49（因为 7^2&#x3D;49 </span><br><span class="line">然后下一个数字是 97（因为 4^2+9^2&#x3D;97) </span><br><span class="line">）。</span><br><span class="line">我们可以不断重复该的过程，直到我们得到 1。因为我们得到了 1，我们知道 7 是一个快乐数，函数应该返回 true。</span><br></pre></td></tr></table></figure>



<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200609/100208586.png" alt="mark"></p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><span class="line">再举一个例子，从116开始。</span><br><span class="line">通过反复平方和计算下一个数字，我们最终得到了58，</span><br><span class="line">在继续计算之后，我们又回到了58.由于我们回到了一个已经计算过的数字，可以知道有一个环，因此不可能达到1。</span><br><span class="line">所以对于116，函数应该返回false</span><br></pre></td></tr></table></figure>

<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200609/100356018.png" alt="mark"></p>
<ul>
<li><strong>思考</strong></li>
</ul>
<p>根据我们的探索，我们猜测会有以下三种可能。</p>
<ol>
<li>最终会得到 1。</li>
<li>最终会进入循环。</li>
<li>值会越来越大，最后接近无穷大。</li>
</ol>
<p>第三个情况比较难以检测和处理。我们怎么知道它会继续变大，而不是最终得到 1 呢？我们可以仔细想一想，每一位数的最大数字的下一位数是多少。</p>
<table>
<thead>
<tr>
<th>Digits</th>
<th align="center">Largest</th>
<th align="right">Next</th>
</tr>
</thead>
<tbody><tr>
<td>1</td>
<td align="center">9</td>
<td align="right">81</td>
</tr>
<tr>
<td>2</td>
<td align="center">99</td>
<td align="right">162</td>
</tr>
<tr>
<td>3</td>
<td align="center">999</td>
<td align="right">243</td>
</tr>
<tr>
<td>4</td>
<td align="center">9999</td>
<td align="right">324</td>
</tr>
<tr>
<td>13</td>
<td align="center">9999999999999</td>
<td align="right">1053</td>
</tr>
</tbody></table>
<ol>
<li>对于三位数的数字，他不可能大于243。</li>
</ol>
<p>（这意味着它要么被困在243以下的循环内，要么跌倒1。）</p>
<ol start="2">
<li>对于4位或者以上的是数字，在每一步都会丢失一位，直到降到3位位置。所以我们知道，最坏的情况下，算法可能会在243以下的循环内或者回到1。但它不会无限期的进行下去，所以我们排除第三种选择。</li>
</ol>
<ul>
<li><strong>算法</strong><ul>
<li>给定一个数字n，判断他的下一个平方后数字是什么</li>
<li>按照一系列数字来判断我们是否进入了一个循环</li>
</ul>
</li>
</ul>
<p>第 1 部分我们按照题目的要求做数位分离，求平方和。</p>
<p>第 2 部分可以使用 HashSet 完成。每次生成链中的下一个数字时，我们都会检查它是否已经在 HashSet 中。</p>
<ul>
<li>如果不在hashset中，把它加入到hashset中</li>
<li>如果在hashset中，说明我们进入了一个循环，返回false。</li>
</ul>
<figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><span class="line"><span class="class"><span class="keyword">class</span> <span class="title">Solution</span> </span>&#123;</span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">boolean</span> <span class="title">isHappy</span><span class="params">(<span class="keyword">int</span> n)</span> </span>&#123;</span><br><span class="line">        Set&lt;Integer&gt; set = <span class="keyword">new</span> HashSet&lt;&gt;();</span><br><span class="line"></span><br><span class="line">        <span class="comment">// 循环条件： 回到了1表示快乐数字 并且 这个数字之前没有出现过</span></span><br><span class="line">        <span class="keyword">while</span> (n != <span class="number">1</span> &amp;&amp; !set.contains(n))&#123;</span><br><span class="line">            <span class="comment">// 加入到set中</span></span><br><span class="line">            set.add(n);</span><br><span class="line">            <span class="comment">// n的下一个平方数</span></span><br><span class="line">            n = getNext(n);</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="comment">// 判断结果</span></span><br><span class="line">        <span class="keyword">return</span> n == <span class="number">1</span>;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="comment">// 计算n 的平方数函数</span></span><br><span class="line">    <span class="function"><span class="keyword">private</span> <span class="keyword">int</span> <span class="title">getNext</span><span class="params">(<span class="keyword">int</span> n)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> total = <span class="number">0</span>;</span><br><span class="line"></span><br><span class="line">        <span class="keyword">while</span> (n &gt; <span class="number">0</span>)&#123;</span><br><span class="line">            <span class="keyword">int</span> d = n % <span class="number">10</span>;</span><br><span class="line">            n = n / <span class="number">10</span>;</span><br><span class="line">            total += d*d;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> total;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>



<p>复杂度分析：（这里参考官网）</p>
<figure class="highlight plain"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line">确定这个问题的时间复杂度对于一个 “简单” 级别的问题来说是一个挑战。如果您对这些问题还不熟悉，可以尝试只计算 getNext(n) 函数的时间复杂度。</span><br><span class="line"></span><br><span class="line">时间复杂度：O(243 \cdot 3 + \log n + \log\log n + \log\log\log n)...O(243⋅3+logn+loglogn+logloglogn)... &#x3D; O(logn)。</span><br><span class="line">查找给定数字的下一个值的成本为 O(logn)，因为我们正在处理数字中的每位数字，而数字中的位数由 \log nlogn 给定。</span><br><span class="line">要计算出总的时间复杂度，我们需要仔细考虑循环中有多少个数字，它们有多大。</span><br><span class="line">我们在上面确定，一旦一个数字低于 243243，它就不可能回到 243243 以上。因此，我们就可以用 243243 以下最长循环的长度来代替 243243，不过，因为常数无论如何都无关紧要，所以我们不会担心它。</span><br><span class="line">对于高于 243243 的 nn，我们需要考虑循环中每个数高于 243243 的成本。通过数学运算，我们可以证明在最坏的情况下，这些成本将是 O(\log n) + O(\log \log n) + O(\log \log \log n)...O(logn)+O(loglogn)+O(logloglogn)...。幸运的是，O(\log n)O(logn) 是占主导地位的部分，而其他部分相比之下都很小（总的来说，它们的总和小于\log nlogn），所以我们可以忽略它们。</span><br><span class="line">空间复杂度：O(\log n)O(logn)。与时间复杂度密切相关的是衡量我们放入 HashSet 中的数字以及它们有多大的指标。对于足够大的 nn，大部分空间将由 nn 本身占用。我们可以很容易地优化到 O(243 \cdot 3) &#x3D; O(1)O(243⋅3)&#x3D;O(1)，方法是只保存集合中小于 243243 的数字，因为对于较高的数字，无论如何都不可能返回到它们。</span><br></pre></td></tr></table></figure>





<h2 id="方法二：快慢指针"><a href="#方法二：快慢指针" class="headerlink" title="方法二：快慢指针"></a>方法二：快慢指针</h2><ul>
<li><p>通过反复调用<code>getNext(n)</code> 得到的是一个<strong>隐式的链表</strong>。</p>
</li>
<li><p>隐式意味着我们没有实际的链表节点和指针，但数据仍然形成链表结构。</p>
</li>
</ul>
<p><strong>如果这里意识到实际上有个链表，那么可以转换成链表是否有环的问题</strong></p>
<ul>
<li><p>因此我们在这里可以使用弗洛伊德循环查找算法。这个算法是两个奔跑选手，一个跑的快，一个跑得慢。在龟兔赛跑的寓言中，跑的快的称为 “乌龟”，跑得快的称为 “兔子”。</p>
</li>
<li><p>不管乌龟和兔子在循环中从哪里开始，它们最终都会相遇。这是因为兔子每走一步就向乌龟靠近一个节点（在它们的移动方向上）。</p>
</li>
</ul>
<p><img src="http://zhuuu-bucket.oss-cn-beijing.aliyuncs.com/img/20200609/111050456.png" alt="mark"></p>
<p><strong>算法：</strong></p>
<ul>
<li>我们不是每次链表中的一个值，而是跟踪两个值，称为快跑者和慢跑者。</li>
<li>在每一步的算法中，慢速在链表中前进一个节点，快跑这前进两个节点。</li>
</ul>
<figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line"></span><br><span class="line"></span><br></pre></td></tr></table></figure>


      
      <!-- reward -->
      
      <div id="reward-btn">
        打赏
      </div>
      
    </div>
      <!-- copyright -->
      
        <div class="declare">
          <ul class="post-copyright">
            <li>
              <i class="ri-copyright-line"></i>
              <strong>版权声明： </strong s>
              本博客所有文章除特别声明外，均采用 <a href="https://www.apache.org/licenses/LICENSE-2.0.html" rel="external nofollow"
                target="_blank">Apache License 2.0</a> 许可协议。转载请注明出处！
            </li>
          </ul>
        </div>
        
    <footer class="article-footer">
      
          
<div class="share-btn">
      <span class="share-sns share-outer">
        <i class="ri-share-forward-line"></i>
        分享
      </span>
      <div class="share-wrap">
        <i class="arrow"></i>
        <div class="share-icons">
          
          <a class="weibo share-sns" href="javascript:;" data-type="weibo">
            <i class="ri-weibo-fill"></i>
          </a>
          <a class="weixin share-sns wxFab" href="javascript:;" data-type="weixin">
            <i class="ri-wechat-fill"></i>
          </a>
          <a class="qq share-sns" href="javascript:;" data-type="qq">
            <i class="ri-qq-fill"></i>
          </a>
          <a class="douban share-sns" href="javascript:;" data-type="douban">
            <i class="ri-douban-line"></i>
          </a>
          <!-- <a class="qzone share-sns" href="javascript:;" data-type="qzone">
            <i class="icon icon-qzone"></i>
          </a> -->
          
          <a class="facebook share-sns" href="javascript:;" data-type="facebook">
            <i class="ri-facebook-circle-fill"></i>
          </a>
          <a class="twitter share-sns" href="javascript:;" data-type="twitter">
            <i class="ri-twitter-fill"></i>
          </a>
          <a class="google share-sns" href="javascript:;" data-type="google">
            <i class="ri-google-fill"></i>
          </a>
        </div>
      </div>
</div>

<div class="wx-share-modal">
    <a class="modal-close" href="javascript:;"><i class="ri-close-circle-line"></i></a>
    <p>扫一扫，分享到微信</p>
    <div class="wx-qrcode">
      <img src="//api.qrserver.com/v1/create-qr-code/?size=150x150&data=http://zhuuu.work/2020/06/08/Leetcode/Leetcode-202-%E5%BF%AB%E4%B9%90%E6%95%B0/" alt="微信分享二维码">
    </div>
</div>

<div id="share-mask"></div>
      
      
  <ul class="article-tag-list" itemprop="keywords"><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/Leetcode/" rel="tag">Leetcode</a></li><li class="article-tag-list-item"><a class="article-tag-list-link" href="/tags/%E5%93%88%E5%B8%8C%E8%A1%A8/" rel="tag">哈希表</a></li></ul>


    </footer>

  </div>

  
  
  <nav class="article-nav">
    
      <a href="/2020/06/09/Leetcode/Leetcode-009-%E5%9B%9E%E6%96%87%E6%95%B0%E5%8F%8D%E8%BD%AC/" class="article-nav-link">
        <strong class="article-nav-caption">上一篇</strong>
        <div class="article-nav-title">
          
            Leetcode-009-回文数反转
          
        </div>
      </a>
    
    
      <a href="/2020/06/07/Leetcode/Leetcode-108-%E5%B0%86%E6%9C%89%E5%BA%8F%E6%95%B0%E7%BB%84%E8%BD%AC%E6%8D%A2%E4%B8%BA%E4%BA%8C%E5%8F%89%E6%90%9C%E7%B4%A2%E6%A0%91/" class="article-nav-link">
        <strong class="article-nav-caption">下一篇</strong>
        <div class="article-nav-title">Leetcode-108-将有序数组转换为二叉搜索树</div>
      </a>
    
  </nav>


  

  
  
<!-- valine评论 -->
<div id="vcomments-box">
    <div id="vcomments">
    </div>
</div>
<script src="//cdn1.lncld.net/static/js/3.0.4/av-min.js"></script>
<script src='https://cdn.jsdelivr.net/npm/valine@1.3.10/dist/Valine.min.js'></script>
<script>
    new Valine({
        el: '#vcomments',
        notify: false,
        verify: '',
        app_id: '',
        app_key: '',
        path: window.location.pathname,
        avatar: 'mp',
        placeholder: '给我的文章加点评论吧~',
        recordIP: true
    });
    const infoEle = document.querySelector('#vcomments .info');
    if (infoEle && infoEle.childNodes && infoEle.childNodes.length > 0) {
        infoEle.childNodes.forEach(function (item) {
            item.parentNode.removeChild(item);
        });
    }
</script>
<style>
    #vcomments-box {
        padding: 5px 30px;
    }

    @media screen and (max-width: 800px) {
        #vcomments-box {
            padding: 5px 0px;
        }
    }

    #vcomments-box #vcomments {
        background-color: #fff;
    }

    .v .vlist .vcard .vh {
        padding-right: 20px;
    }

    .v .vlist .vcard {
        padding-left: 10px;
    }
</style>

  

  
  
<div class="gitalk" id="gitalk-container"></div>

<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/gitalk@1.5.0/dist/gitalk.css">


<script src="https://cdn.jsdelivr.net/npm/gitalk@1.5.0/dist/gitalk.min.js"></script>


<script src="https://cdn.jsdelivr.net/npm/blueimp-md5@2.10.0/js/md5.min.js"></script>

<script type="text/javascript">
  var gitalk = new Gitalk({
    clientID: 'db188ed8c86dc4b0dbf3',
    clientSecret: 'a58f92160e5a9efd726b7d533000a0737f3e3f3e',
    repo: 'Blog-comments',
    owner: 'Zhuuuuuuuu',
    admin: ['Zhuuuuuuuu'],
    // id: location.pathname,      // Ensure uniqueness and length less than 50
    id: md5(location.pathname),
    distractionFreeMode: false,  // Facebook-like distraction free mode
    pagerDirection: 'last'
  })

  gitalk.render('gitalk-container')
</script>

  

</article>
</section>
      <footer class="footer">
  <div class="outer">
    <ul class="list-inline">
      <li>
        &copy;
        2019-2021
        Zhuuu
      </li>
      <li>
        
      </li>
    </ul>
    <ul class="list-inline">
      <li>
        
        
        <span>
  <i>PV:<span id="busuanzi_value_page_pv"></span></i>
  <i>UV:<span id="busuanzi_value_site_uv"></span></i>
</span>
        
      </li>
      <li>
        <!-- cnzz统计 -->
        
        <script type="text/javascript" src='https://s9.cnzz.com/z_stat.php?id=1278069914&amp;web_id=1278069914'></script>
        
      </li>
    </ul>
  </div>
</footer>
    <div class="to_top">
        <div class="totop" id="totop">
  <i class="ri-arrow-up-line"></i>
</div>
      </div>
    </main>
      <aside class="sidebar">
        <button class="navbar-toggle"></button>
<nav class="navbar">
  
  <div class="logo">
    <a href="/"><img src="/images/ayer-side.svg" alt="朱酱酱的学习博客"></a>
  </div>
  
  <ul class="nav nav-main">
    
    <li class="nav-item">
      <a class="nav-item-link" href="/">主页</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags">标签</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/JVM/">JVM</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/JDK%E6%BA%90%E7%A0%81%E5%88%86%E6%9E%90/">JDK源码</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/%E5%A4%9A%E7%BA%BF%E7%A8%8B/">多线程</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/Mysql/">Mysql</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/Redis/">Redis</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/%E8%AE%BE%E8%AE%A1%E8%80%85%E6%A8%A1%E5%BC%8F/">设计模式</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/MyBatis/">MyBatis</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/SpringMVC/">SpringMVC</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/Spring/">Spring</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/SpringBoot/">SpringBoot</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/%E6%93%8D%E4%BD%9C%E7%B3%BB%E7%BB%9F/">Linux</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/Leetcode/">Leetcode</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/%E5%89%8D%E7%AB%AF/">前端</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/%E7%BD%91%E7%BB%9C%E7%BC%96%E7%A8%8B/">网络编程</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/tags/photoshop/">photoshop</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="http://smartzhuuu.lofter.com/" target="_blank" rel="noopener">摄影</a>
    </li>
    
    <li class="nav-item">
      <a class="nav-item-link" href="/2020/about">关于我</a>
    </li>
    
  </ul>
</nav>
<nav class="navbar navbar-bottom">
  <ul class="nav">
    <li class="nav-item">
      
      <a class="nav-item-link nav-item-search"  title="Search">
        <i class="ri-search-line"></i>
      </a>
      
      
      <a class="nav-item-link" target="_blank" href="/atom.xml" title="RSS Feed">
        <i class="ri-rss-line"></i>
      </a>
      
    </li>
  </ul>
</nav>
<div class="search-form-wrap">
  <div class="local-search local-search-plugin">
  <input type="search" id="local-search-input" class="local-search-input" placeholder="Search...">
  <div id="local-search-result" class="local-search-result"></div>
</div>
</div>
      </aside>
      <div id="mask"></div>

<!-- #reward -->
<div id="reward">
  <span class="close"><i class="ri-close-line"></i></span>
  <p class="reward-p"><i class="ri-cup-line"></i>请我喝杯咖啡吧~</p>
  <div class="reward-box">
    
    <div class="reward-item">
      <img class="reward-img" src="/images/alipay.jpg">
      <span class="reward-type">支付宝</span>
    </div>
    
    
    <div class="reward-item">
      <img class="reward-img" src="/images/wechat.jpg">
      <span class="reward-type">微信</span>
    </div>
    
  </div>
</div>
      
<script src="/js/jquery-2.0.3.min.js"></script>


<script src="/js/jquery.justifiedGallery.min.js"></script>


<script src="/js/lazyload.min.js"></script>


<script src="/js/busuanzi-2.3.pure.min.js"></script>


<script src="/js/share.js"></script>



<script src="/fancybox/jquery.fancybox.min.js"></script>




<script>
  try {
    var typed = new Typed("#subtitle", {
    strings: ['昨夜西风凋碧树。独上高楼，望尽天涯路','衣带渐宽终不悔，为伊消得人憔悴。','众里寻他千百度。蓦然回首，那人却在，灯火阑珊处。'],
    startDelay: 0,
    typeSpeed: 200,
    loop: true,
    backSpeed: 100,
    showCursor: true
    });
  } catch (err) {
  }
  
</script>




<script src="/js/tocbot.min.js"></script>

<script>
  // Tocbot_v4.7.0  http://tscanlin.github.io/tocbot/
  tocbot.init({
    tocSelector: '.tocbot',
    contentSelector: '.article-entry',
    headingSelector: 'h1, h2, h3, h4, h5, h6',
    hasInnerContainers: true,
    scrollSmooth: true,
    scrollContainer:'main',
    positionFixedSelector: '.tocbot',
    positionFixedClass: 'is-position-fixed',
    fixedSidebarOffset: 'auto',
    onClick: (e) => {
      $('.toc-link').removeClass('is-active-link');
      $(`a[href=${e.target.hash}]`).addClass('is-active-link');
      $(e.target.hash).scrollIntoView();
      return false;
    }
  });
</script>


<script>
  var ayerConfig = {
    mathjax: true
  }
</script>


<script src="/js/ayer.js"></script>


<script src="https://cdn.jsdelivr.net/npm/jquery-modal@0.9.2/jquery.modal.min.js"></script>
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/jquery-modal@0.9.2/jquery.modal.min.css">


<!-- Root element of PhotoSwipe. Must have class pswp. -->
<div class="pswp" tabindex="-1" role="dialog" aria-hidden="true">

    <!-- Background of PhotoSwipe. 
         It's a separate element as animating opacity is faster than rgba(). -->
    <div class="pswp__bg"></div>

    <!-- Slides wrapper with overflow:hidden. -->
    <div class="pswp__scroll-wrap">

        <!-- Container that holds slides. 
            PhotoSwipe keeps only 3 of them in the DOM to save memory.
            Don't modify these 3 pswp__item elements, data is added later on. -->
        <div class="pswp__container">
            <div class="pswp__item"></div>
            <div class="pswp__item"></div>
            <div class="pswp__item"></div>
        </div>

        <!-- Default (PhotoSwipeUI_Default) interface on top of sliding area. Can be changed. -->
        <div class="pswp__ui pswp__ui--hidden">

            <div class="pswp__top-bar">

                <!--  Controls are self-explanatory. Order can be changed. -->

                <div class="pswp__counter"></div>

                <button class="pswp__button pswp__button--close" title="Close (Esc)"></button>

                <button class="pswp__button pswp__button--share" style="display:none" title="Share"></button>

                <button class="pswp__button pswp__button--fs" title="Toggle fullscreen"></button>

                <button class="pswp__button pswp__button--zoom" title="Zoom in/out"></button>

                <!-- Preloader demo http://codepen.io/dimsemenov/pen/yyBWoR -->
                <!-- element will get class pswp__preloader--active when preloader is running -->
                <div class="pswp__preloader">
                    <div class="pswp__preloader__icn">
                        <div class="pswp__preloader__cut">
                            <div class="pswp__preloader__donut"></div>
                        </div>
                    </div>
                </div>
            </div>

            <div class="pswp__share-modal pswp__share-modal--hidden pswp__single-tap">
                <div class="pswp__share-tooltip"></div>
            </div>

            <button class="pswp__button pswp__button--arrow--left" title="Previous (arrow left)">
            </button>

            <button class="pswp__button pswp__button--arrow--right" title="Next (arrow right)">
            </button>

            <div class="pswp__caption">
                <div class="pswp__caption__center"></div>
            </div>

        </div>

    </div>

</div>

<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/photoswipe.min.css">
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/default-skin/default-skin.css">
<script src="https://cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/photoswipe.min.js"></script>
<script src="https://cdn.jsdelivr.net/npm/photoswipe@4.1.3/dist/photoswipe-ui-default.min.js"></script>

<script>
    function viewer_init() {
        let pswpElement = document.querySelectorAll('.pswp')[0];
        let $imgArr = document.querySelectorAll(('.article-entry img:not(.reward-img)'))

        $imgArr.forEach(($em, i) => {
            $em.onclick = () => {
                // slider展开状态
                // todo: 这样不好，后面改成状态
                if (document.querySelector('.left-col.show')) return
                let items = []
                $imgArr.forEach(($em2, i2) => {
                    let img = $em2.getAttribute('data-idx', i2)
                    let src = $em2.getAttribute('data-target') || $em2.getAttribute('src')
                    let title = $em2.getAttribute('alt')
                    // 获得原图尺寸
                    const image = new Image()
                    image.src = src
                    items.push({
                        src: src,
                        w: image.width || $em2.width,
                        h: image.height || $em2.height,
                        title: title
                    })
                })
                var gallery = new PhotoSwipe(pswpElement, PhotoSwipeUI_Default, items, {
                    index: parseInt(i)
                });
                gallery.init()
            }
        })
    }
    viewer_init()
</script>



<script type="text/x-mathjax-config">
  MathJax.Hub.Config({
      tex2jax: {
          inlineMath: [ ['$','$'], ["\\(","\\)"]  ],
          processEscapes: true,
          skipTags: ['script', 'noscript', 'style', 'textarea', 'pre', 'code']
      }
  });

  MathJax.Hub.Queue(function() {
      var all = MathJax.Hub.getAllJax(), i;
      for(i=0; i < all.length; i += 1) {
          all[i].SourceElement().parentNode.className += ' has-jax';
      }
  });
</script>

<script src="https://cdn.jsdelivr.net/npm/mathjax@2.7.6/unpacked/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>


<script type="text/javascript" src="https://js.users.51.la/20544303.js"></script>
  </div>
</body>

</html>